Stability and distortions of liquid crystal order in a cell with a heterogeneous substrate

We study stability and distortions of liquid crystal nematic order in a cell with a random heterogeneous substrate. Modeling this system as a bulk xy model with quenched disorder confined to a surface, we find that nematic order is marginally unstable to such surface pinning. We compute the length scale beyond which nematic distortions become large and calculate orientational correlation functions using the functional renormalization-group and matching methods, finding universal logarithmic and double-logarithmic distortions in two and three dimensions, respectively. We extend these results to a finite-thickness liquid crystal cell with a second homogeneous substrate, detailing crossovers as a function of random pinning strength and cell thickness. We conclude with analysis of experimental signatures of these distortions in a conventional crossed-polarizer-analyzer light microscopy.